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Rigidity of Riemannian manifolds with vanishing generalized Bach tensor
Journal of Geometry and Physics ( IF 1.6 ) Pub Date : 2021-09-23 , DOI: 10.1016/j.geomphys.2021.104380
Guangyue Huang 1 , Bingqing Ma 1 , Xingxiao Li 1
Affiliation  

A generalized Bach tensor Bijt with parameter tR is introduced. A Riemannian manifold (Mn,g) is called Bt-flat if its generalised Bach tensor Bijt0 for some parameter t. In this paper, we first study the rigidity of closed Bt-flat Riemannian manifolds with positive constant scalar curvature. When the dimension n=4, we prove that all Bt-flat manifolds that satisfy a point-wise inequality must be of positive constant sectional curvature. Similar rigidity results are also obtained in terms of the Yamabe invariant. Moreover, using the curvature estimates for the general dimension n4, we obtain an integral inequality for Bt-flat manifolds, and prove that the equality occurs if and only if these manifolds are of positive constant sectional curvature. In addition, we also obtain similar rigidity results for complete Bt-flat manifolds.



中文翻译:

具有消失广义巴赫张量的黎曼流形的刚性

广义巴赫张量 一世j 带参数 电阻介绍。黎曼流形(n,G) 叫做 -flat 如果它的广义巴赫张量 一世j0对于某些参数t。在本文中,我们首先研究封闭的刚性-具有正常数标量曲率的平坦黎曼流形。当维度n=4,我们证明所有 -满足逐点不等式的平坦流形必须具有正的恒定截面曲率。在 Yamabe 不变量方面也获得了类似的刚性结果。此外,使用一般维度的曲率估计n4,我们得到一个积分不等式 -扁平流形,并证明当且仅当这些流形具有正的恒定截面曲率时,等式才会发生。此外,我们还获得了完全相似的刚性结果-扁平歧管。

更新日期:2021-10-04
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